Third quantization

TL;DR

This paper extends the concepts of second quantization and Fock space to density operator states for open many-body systems, focusing on bosonic cases and their applications to non-equilibrium steady states.

Contribution

It introduces a third quantization framework for density operators, including dual space constructions for bosons, and discusses applications to non-equilibrium steady states.

Findings

Developed a third quantization formalism for density operators.

Constructed dual spaces for bosonic operators to handle unboundedness.

Applied the framework to analyze non-equilibrium steady states.

Abstract

The basic ideas of second quantization and Fock space are extended to density operator states, used in treatments of open many-body systems. This can be done for fermions and bosons. While the former only requires the use of a non-orthogonal basis, the latter requires the introduction of a dual set of spaces. In both cases an operator algebra closely resembling the canonical one is developed and used to define the dual sets of bases. We here concentrated on the bosonic case where the unboundedness of the operators requires the definitions of dual spaces to support the pair of bases. Some applications, mainly to non-equilibrium steady states, will be mentioned.